OFFSET
3,2
LINKS
Peter Luschny, The difference of the Stirling cycle numbers and the Stirling set numbers, Mathematics Stack Exchange, Feb. 2021.
FORMULA
T(n, k) = Sum_{j=0..k} (binomial(n+j-1, 2*k) - binomial(n+k-j, 2*k))*A340556(k, j).
E.g.f.: (1 - z)^(-x) - exp(x*(exp(z) - 1)) (unrestricted rows and n >= 0).
EXAMPLE
Triangle starts:
[ 3] [1]
[ 4] [5, 4]
[ 5] [23, 35, 10]
[ 6] [119, 243, 135, 20]
[ 7] [719, 1701, 1323, 385, 35]
[ 8] [5039, 12941, 12166, 5068, 910, 56]
[ 9] [40319, 109329, 115099, 59514, 15498, 1890, 84]
[10] [362879, 1026065, 1163370, 689575, 226800, 40446, 3570, 120]
MAPLE
# Giving full rows for n >= 0:
gf := (1 - z)^(-x) - exp(x*(exp(z) - 1));
ser := series(gf, z, 20): coeffz := n -> coeff(ser, z, n):
A341102row := n -> seq(n!*coeff(coeffz(n), x, k), k=0..n):
for n from 0 to 9 do A341102row(n) od;
PROG
(SageMath)
for n in (3..11):
print([stirling_number1(n, k) - stirling_number2(n, k) for k in (1..n-2)])
(PARI) T(n, k) = abs(stirling(n, k, 1)) - stirling(n, k, 2); \\ Michel Marcus, Feb 24 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Feb 24 2021
STATUS
approved